# A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Dynamic Multi-Objective Problem

**Definition**

**1.**

**.**For some time t, if individual p, it can as $p\prec q$. If and only if $\forall \text{}i=\{1,2,\dots ,m\}$: ${f}_{i}(p,t)\le {f}_{i}(q,t)$, $\exists \text{}\mathrm{j}=\{1,2,\dots ,m\}$:${f}_{j}(p,t)<{f}_{j}(q,t)$.

**Definition**

**2.**

**.**Let $x\in \mathsf{\Omega}$, which is the decision variable. PS could be defined as follows,

**Definition**

**3.**

**.**Let $x\in \mathsf{\Omega}$, which is the decision variable. PF could be defined as follows,

## 3. Segmentation Cloud Prediction Strategy

#### 3.1. Population Segmentation

^{2}). Therefore, the proposed method of segmentation is simpler for population segmentation and has lower computation complexity.

#### 3.2. Directional Cloud Prediction Strategy

_{i}contains K

_{i}individuals. The center of sub-population Pop

_{i}could be described as follows,

**d**

_{i}(t) and the predicted error of two migrations $\mathbf{\Delta}\left(t\right)$ at time t+1 could be calculated as follows, based on the predictions of center positions of populations at time t and time t-1.

**v**

_{1}, and the individual position of prediction after change is y.

#### 3.3. Extra Angle Search

_{1}*, …, En

_{n}*). Then, the normal random vector ${v}_{2}~N\left({\mathit{e}}_{i}\right(t),En{*}^{2})$ can be generated, whose expectation is ${\mathit{e}}_{i}\left(t\right)$ and standard deviation is $En*$. The position of prediction from the angular deviation search can be calculated as follow,

_{2}, and the position of prediction after change is y*.

#### 3.4. Environmental Detection

#### 3.5. SCPS Framework

**Input:**When enough historical information cannot be collected, the proportion of the population in random initialization is $\zeta $, the proportion of directional prediction population is L

_{1}, the population size is N, and the final time of environmental change is T

_{max}, t:= 0, d(t − 1):= 0.

**Output:**PS.

**Step 1:**Randomly initialize the population.

**Step 2:**According to Equation (11), detect if the environment changes or not. If change, turn to Step 3; otherwise, turn to Step 8.

**Step 3:**If the value of d(t − 1) is 0, turn to Step 4; otherwise turn to Step 5.

**Step 4:**Randomly select $\zeta $ $\times $ N individuals to evolve, let d(t − 1) = d(t).

**Step 5:**According to Section 1, segment the population into m + 1 parts. Calculate the center of the population by Equation (4), and calculate the moving direction of population at time t. Then select L

_{1}× N individuals with the binary tournament selection model and predict the position with the cloud model.

**Step 6:**Select N$\times $(1 − L

_{1}) individuals with the binary tournament selection model and calculate the search vector of extra angle search for these individuals using Equation (7) to Equation (9).

**Step 7:**Boundary detect for new individuals.

**Step 8:**Calculate the non-dominated sort and crowding distance. Hold the first n individuals.

**Step 9:**Termination conditional judgment. If meet, turn to Step 10. Otherwise turn to Step 2.

**Step 10:**End, output the population PS.

_{1}$\times $ N and N $\times $ (1 − K

_{1}) individuals to predict, respectively. As the predicted individuals might be beyond the decision space, the Step 7 is to boundary detect new individuals, and revise the ones beyond.

## 4. Experimental Analysis

#### 4.1. Benchmark Problems

#### 4.2. Parameter Setting

_{max}= 10.

_{1}= 0.5N.

#### 4.3. Metrics

#### 4.4. Test and Analysis

^{−2}magnitude. That is, all these four algorithms have good ability to track the dynamic front.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Xia, X.; Gui, L.; Zhan, Z. A multi-swarm particle swarm optimization algorithm based on dynamical topology and purposeful detecting. Appl. Soft Comput.
**2018**, 67, 126–140. [Google Scholar] [CrossRef] - Song, Y.; Fu, Q.; Wang, Y.; Wang, X. Divergence-based cross entropy and uncertainty measures of Atanassov’s intuitionistic fuzzy sets with their application in decision making. Appl. Soft Comput. J.
**2019**, 84, 105703. [Google Scholar] [CrossRef] - Zhang, B.; Zhang, M.; Song, Y.; Zhang, L. Combing evidence sources in time domain with decision maker’s preference on time sequence. IEEE Access
**2019**, 7, 174210–174218. [Google Scholar] [CrossRef] - Song, Y.; Wang, X.; Quan, W.; Huang, W. A new approach to construct similarity measure for intuitionistic fuzzy sets. Soft Comput.
**2019**, 23, 1985–1998. [Google Scholar] [CrossRef] - Song, Y.; Wang, X.; Zhu, J.; Lei, L. Sensor dynamic reliability evaluation based on evidence and intuitionistic fuzzy sets. Appl. Intell.
**2018**, 48, 3950–3962. [Google Scholar] [CrossRef] - Song, Y.; Wang, X.; Lei, L.; Xue, A. A novel similarity measure on intuitionistic fuzzy sets with its applications. Appl. Intell.
**2015**, 42, 252–261. [Google Scholar] [CrossRef] - Lei, L.; Song, Y.; Luo, X. A New Re-encoding ECOC Using a Reject Option. Appl. Intell. [CrossRef]
- Soh, H.; Ong, Y.S.; Nguyen, Q.C.; Nguyen, Q.H.; Habibullah, M.S.; Hung, T.; Kuo, J.L. Discovering unique, low-energy pure water isomers: memetic exploration, optimization and landscape analysis. IEEE Trans. Comput.
**2010**, 14, 419–437. [Google Scholar] [CrossRef] - Thammawichai, M.; Kerrigan, E.C. Energy-efficient real-time scheduling for two-type heterogeneous multiprocessors. Real Time Syst.
**2018**, 54, 132–165. [Google Scholar] [CrossRef] [Green Version] - Shen, M.; Zhan, Z.H.; Chen, W.N.; Gong, Y.J.; Zhang, J.; Li, Y. Bi-velocity discrete particle swarm optimization and its application to multicast routing problem in communication networks. IEEE Trans. Ind. Electron.
**2014**, 61, 7141–7151. [Google Scholar] [CrossRef] - Deb, K.; Karthik, S. Dynamic multiobjective optimization and decision-making using modified NSGA-II: A case study on hydro-thermal power scheduling. In Proceedings of the 4th International Conference on Evolutionary Multi-Criterion Optimization, Matsushima, Japan, 5–8 March 2007; Springer: Berlin/Heidelberg, Germany, 2007; pp. 803–817. [Google Scholar]
- Azevedo, C.R.B.; Araujo, A.F.R. Generalized immigration schemes for dynamic evolutionary multiobjective optimization. In Proceedings of the IEEE Congress on Evolutionary Computation, New Orleans, LA, USA, 5–8 June 2011; pp. 2033–2040. [Google Scholar]
- Shang, R.H.; Jiao, L.C.; Gong, M.G.; Ma, W.P. An immune clonal algorithm for dynamic multi-objective optimization. J. Softw.
**2007**, 18, 2700–2711. [Google Scholar] [CrossRef] - Kominami, M.; Hamagami, T. A new genetic algorithm with diploid chromosomes by using probability decoding for adaptation to various environments. Electron. Commun. Jpn.
**2010**, 93, 38–46. [Google Scholar] [CrossRef] - Yang, S. On the Design of Diploid Genetic Algorithms for Problem Optimization in Dynamic Environments. In Proceedings of the 2006 Congress on Evolutionary Computation, Vancouver, BD, Canada, 16–21 July 2006; pp. 1362–1369. [Google Scholar]
- Liu, M.; Zeng, W.H. Memory enhanced dynamic multi-objective evolutionary algorithm based on decomposition. J. Softw.
**2013**, 24, 1571–1588. [Google Scholar] [CrossRef] - Hatzakis, I.; Wallace, D. Dynamic multi-objective optimization with evolutionary algorithms: A forward-looking approach. In Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, Washington, DC, USA, 8–12 July 2006; pp. 1201–1208. [Google Scholar]
- Zhou, A.M.; Jin, Y.C.; Zhang, Q.F. A population prediction strategy for evolutionary dynamic multiobjective optimization. IEEE Trans. Cybern.
**2014**, 44, 40–53. [Google Scholar] [CrossRef] - Wu, Y.; Jin, Y.; Liu, X. A directed search strategy for evolutionary dynamic multiobjetive optimization. Soft Comput.
**2015**, 19, 3221–3235. [Google Scholar] [CrossRef] - Rong, M.; Gong, D.; Zhang, Y.; Jin, Y.; Pedrycz, W. Multidirectional prediction approach for dynamic multiobjective optimization problems. In Intelligent Computing Methodologies, ICIC 2016 Lecture Notes in Computer Science; Springer: Cham, Switzerland, 2016; Volume 9773. [Google Scholar]
- Li, Q.; Zou, J.; Yang, S.; Zheng, J.; Ruan, G. A predictive strategy based on special points for evolutionary dynamic multi-objective optimization. Soft Comput.
**2018**, 1–17. [Google Scholar] [CrossRef] [Green Version] - Ruan, G.; Yu, G.; Zheng, J.; Zou, J.; Yang, S. The effect of diversity maintenance on prediction in dynamic multi-objective optimization. Appl. Soft Comput.
**2017**, 56, 631–647. [Google Scholar] [CrossRef] - Gee, S.B.; Tan, K.C.; Abbass, H.A. A benchmark test suite for dynamic evolutionary multiobjective optimization. IEEE Trans. Cybern.
**2017**, 47, 461–472. [Google Scholar] [CrossRef] - Deb, K.; Agrawal, S.; Pratap, A.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] [Green Version] - Jiang, S.; Yang, S. Evolutionary dynamic multiobjective optimization: bencmarks and algorithm comparisons. IEEE Trans. Cybern.
**2017**, 47, 198–211. [Google Scholar] [CrossRef] - Zhang, Q.; Li, H. MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput.
**2007**, 11, 712–731. [Google Scholar] [CrossRef]

Function | n, τ | SCPS | MDP | PSS | MOEAD | ||||
---|---|---|---|---|---|---|---|---|---|

Mean Value | Variance | Mean Value | Variance | Mean Value | Variance | Mean Value | Variance | ||

JY1 | 5,10 | 1.75 × 10^{−2} | 3.99 × 10^{−4} | 4.87 × 10^{−2} | 1.18 × 10^{−3} | 2.03 × 10^{−2} | 4.79 × 10^{−3} | 5.16 × 10^{−2} | 2.06 × 10^{−3} |

10,10 | 7.55 × 10^{−3} | 1.31 × 10^{−4} | 1.64 × 10^{−2} | 3.86 × 10^{−4} | 3.17 × 10^{−2} | 9.98 × 10^{−3} | 3.43 × 10^{−2} | 1.16 × 10^{−3} | |

10,20 | 4.28 × 10^{−3} | 6.01 × 10^{−5} | 8.50 × 10^{−3} | 1.30 × 10^{−4} | 8.73 × 10^{−3} | 1.53 × 10^{−3} | 2.34 × 10^{−2} | 1.77 × 10^{−3} | |

JY2 | 5,10 | 4.76 × 10^{−2} | 2.15 × 10^{−4} | 6.17 × 10^{−2} | 7.78 × 10^{−4} | 5.09 × 10^{−2} | 9.68 × 10^{−4} | 7.15 × 10^{−2} | 1.27 × 10^{−3} |

10,10 | 7.39 × 10^{−3} | 5.10 × 10^{−5} | 1.66 × 10^{−2} | 3.06 × 10^{−4} | 3.09 × 10^{−2} | 1.38 × 10^{−2} | 3.30 × 10^{−2} | 6.96 × 10^{−4} | |

10,20 | 4.13 × 10^{−3} | 3.05 × 10^{−5} | 8.30 × 10^{−3} | 1.28 × 10^{−4} | 8.47 × 10^{−3} | 1.27 × 10^{−3} | 2.48 × 10^{−2} | 1.83 × 10^{−3} | |

JY3 | 5,10 | 9.17 × 10^{−3} | 4.36 × 10^{−4} | 1.75 × 10^{−2} | 3.46 × 10^{−4} | 1.52 × 10^{−1} | 1.33 × 10^{−1} | 2.12 × 10^{−1} | 1.86 × 10^{−1} |

10,10 | 9.11 × 10^{−3} | 6.18 × 10^{−4} | 1.74 × 10^{−2} | 3.07 × 10^{−4} | 2.91 × 10^{−1} | 2.38 × 10^{−1} | 1.87 × 10^{−1} | 1.78 × 10^{−1} | |

10,20 | 5.09 × 10^{−3} | 7.72 × 10^{−5} | 1.34 × 10^{−2} | 1.74 × 10^{−4} | 1.26 × 10^{−2} | 1.60 × 10^{−3} | 1.02 × 10^{−1} | 1.30 × 10^{−1} | |

JY4 | 5,10 | 4.07 × 10^{−2} | 1.05 × 10^{−3} | 8.55 × 10^{−2} | 1.57 × 10^{−3} | 3.85 × 10^{−2} | 5.97 × 10^{−3} | 5.66 × 10^{−2} | 2.03 × 10^{−3} |

10,10 | 2.52 × 10^{−2} | 5.52 × 10^{−4} | 5.36 × 10^{−2} | 5.52 × 10^{−4} | 5.25 × 10^{−2} | 7.97 × 10^{−3} | 3.72 × 10^{−2} | 1.15 × 10^{−3} | |

10,20 | 3.41 × 10^{−3} | 6.64 × 10^{−5} | 5.43 × 10^{−3} | 6.43 × 10^{−5} | 7.47 × 10^{−3} | 1.01 × 10^{−3} | 1.07 × 10^{−2} | 4.64 × 10^{−4} | |

JY5 | 5,10 | 2.42 × 10^{−3} | 5.83 × 10^{−6} | 1.13 × 10^{−2} | 2.38 × 10^{−4} | 1.82 × 10^{−2} | 9.69 × 10^{−3} | 3.48 × 10^{−2} | 6.36 × 10^{−4} |

10,10 | 2.42 × 10^{−3} | 3.29 × 10^{−6} | 1.10 × 10^{−2} | 2.06 × 10^{−4} | 2.11 × 10^{−2} | 1.31 × 10^{−2} | 2.46 × 10^{−2} | 1.03 × 10^{−3} | |

10,20 | 2.41 × 10^{−3} | 7.04 × 10^{−6} | 6.03 × 10^{−3} | 5.48 × 10^{−5} | 7.41 × 10^{−3} | 2.17 × 10^{−3} | 2.06 × 10^{−2} | 1.52 × 10^{−4} | |

JY6 | 5,10 | 2.11 | 6.81 × 10^{−2} | 5.90 | 1.65 × 10^{−1} | 2.97 | 1.96 × 10^{−1} | 3.35 | 1.18 × 10^{−1} |

10,10 | 1.61 | 6.26 × 10^{−2} | 3.34 | 6.99 × 10^{−2} | 3.23 | 3.76 × 10^{−1} | 2.04 | 1.25 × 10^{−1} | |

10,20 | 3.88 × 10^{−1} | 1.34 × 10^{−2} | 1.18 | 3.45 × 10^{−2} | 7.97 × 10^{−1} | 8.59 × 10^{−2} | 5.36 × 10^{−1} | 1.49 × 10^{−2} | |

JY7 | 5,10 | 1.42 | 1.26 × 10^{−1} | 16.6 | 1.68 | 89.8 | 64.7 | 20.51 | 3.76 |

10,10 | 2.06 × 10^{−1} | 9.54 × 10^{−2} | 4.33 × 10^{−1} | 5.27 × 10^{−2} | 83.9 | 67.6 | 8.38 × 10^{−1} | 5.71 × 10^{−1} | |

10,20 | 8.55 × 10^{−2} | 4.93 × 10^{−3} | 2.40 × 10^{−1} | 2.10 × 10^{−2} | 5.80 | 7.09 | 3.35 × 10^{−1} | 8.29 × 10^{−2} | |

JY8 | 5,10 | 3.34 × 10^{−3} | 4.25 × 10^{−5} | 1.12 × 10^{−2} | 2.12 × 10^{−4} | 1.93 × 10^{−2} | 1.38 × 10^{−2} | 6.31 × 10^{−2} | 3.93 × 10^{−3} |

10,10 | 3.17 × 10^{−3} | 3.45 × 10^{−5} | 1.13 × 10^{−2} | 2.23 × 10^{−4} | 2.86 × 10^{−2} | 1.44 × 10^{−2} | 6.34 × 10^{−2} | 1.92 × 10^{−2} | |

10,20 | 2.77 × 10^{−3} | 2.06 × 10^{−5} | 6.72 × 10^{−3} | 1.19 × 10^{−4} | 7.57 × 10^{−3} | 1.93 × 10^{−3} | 4.21 × 10^{−2} | 1.47 × 10^{−3} |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ni, P.; Gao, J.; Song, Y.; Quan, W.; Xing, Q.
A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction. *Symmetry* **2020**, *12*, 465.
https://doi.org/10.3390/sym12030465

**AMA Style**

Ni P, Gao J, Song Y, Quan W, Xing Q.
A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction. *Symmetry*. 2020; 12(3):465.
https://doi.org/10.3390/sym12030465

**Chicago/Turabian Style**

Ni, Peng, Jiale Gao, Yafei Song, Wen Quan, and Qinghua Xing.
2020. "A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction" *Symmetry* 12, no. 3: 465.
https://doi.org/10.3390/sym12030465